Chapter 4. Profit and Bayesian Optimality

Transcription

1 Chpter 4 Profit nd Byesin Optimlity In this chpter we consider the objective of profit. The objective of profit mximiztion dds significnt new chllenge over the previously considered objective of socil surplus mximiztion. Fundmentlly, where for socil surplus there is lwys single optiml mechnism (bsent computtionl constrints), for profit there is no single optiml mechnism. For ny mechnism, there is setting nd nother mechnism where this new mechnism hs strictly lrger profit thn the first one. This non-existence of n bsolutely optiml mechnism requires relxtion of wht we consider good mechnism. In this text we will consider two pproches. This chpter focuses on the trditionl economics pproch of Byesin optimiztion. We will ssume tht the distribution of the gents preferences is common knowledge, even to the mechnism designer. This designer should then serch for the mechnism tht mximizes their expected profit when preferences re indeed drwn from the distribution. In Chpter 6 we will consider the trditionl in computer science pproch. We will look for single mechnism tht is good in ny setting. We know tht it is not possible to be optiml, so insted the designer should look for the mechnism is the best pproximtion to the optiml one, for instnce, to the Byesin optiml one tht follows from the forementioned economics pproch. As in the setting of socil surplus we will be considering generl single-dimensionl gent settings, i.e., ech gent hs single vlue for service nd the designer hs cost function over the sets of served gents (see Section 3.1). The profit of the mechnism with outcome x nd pyments p is Profit(p,x) = i p i c(x) (Definition 3.5). Here the setting is given by c( ) nd nd distribution ssumption we my mke on the vlues F. For generl fesibility settings, where the designer s cost for ny fesible outcome is zero, we will refer to the profit s revenue. A motivting exmple ws given in Chpter 1 with single-item uction with two gents whose vlues re drwn independently nd identiclly from U[, 1]. We clculted the expected revenue of the second-price uction nd determined it ws simply the expecttion of the lower vlue, i.e, 1/3. We then clculted the expected revenue of the second-price uction with reserve price 1/2 nd determined tht it ws 5/12. We concluded tht setting reserve price cn improve the revenue of n uction; however, we did not solve for the optiml uction. We show tht the second-price uction with reserve 1/2 is indeed optiml for 63

2 this two bidder exmple nd furthermore we give concise chrcteriztion of the optiml uction for ny single-dimensionl gent setting. 4.1 Revenue Curves We strt by removing ll the compliction of mechnisms for multiple gents nd consider only single gent, Alice, desiring single item. Suppose Alice s vlue v is drwn from distribution F. How should we sell the item to Alice to mximize our profit? A first step would be to consider offering Alice tke-it-or-leve-it price p. Wht does Alice do with such price? If Alice s vlue v p then Alice buys the item nd pys p, otherwise Alice does not buy the item nd pys zero. Therefore, we cn clculte the expected revenue s p times the probbility tht Alice s vlue exceeds p, which cn be red directly form the density function s 1 F(p). Our revenue s function of p is p (1 F(p)). Definition 4.1 The revenue curve B(p) is the revenue obtined from gent with vlue distribution F from posted price p s function of tht price, i.e., B(p) = p (1 F(p)). We cn clerly optimize this by tking the derivtive nd setting it equl to zero. If F is U[, 1] then F(p) = p nd the revenue is optimized t price p = 1/2 where n expected revenue of 1/4 is chieved. The uniform distribution is well-behved in the sense tht the revenue, s function of price, increses up to the optiml price of 1/2 nd then decreses. The importnce of the derivtive in solving for the optiml price cn be noted by observing tht the derivtive is negtive but incresing s p is lowered to p = 1/2, where it is zero, nd then continues to be positive nd incresing fterwrds. This optiml pricing rule is llocting where the derivtive of this revenue curve is negtive!. 4.2 Expected Revenue nd Virtul Vlues Suppose we re given the lloction rule of n gent (Alice) s x(v). By the pyment identity, the pyment rule must be p(v) = vx(v) 1 x(z)dz. Since v is drwn from F we cn clculte Alice s ex nte expected pyment. [ v ] E v F [p(v)] = vx(v) x(z)dz dv. We could esily nlyze this expression by noting the double integrl, swpping order of integrtion, nd simplifying. However, such tretment, though formuliclly correct, gives less intuition for the relevnt quntities. The following is brief digression into integrtion by prts. Integrte the product rule for differentition, e.g., with g (z) denoting the derivtive of g(z) with respect to z. g(z)h(z) b = b g (z)h(z)dz + 64 b g(z)h (z)dz.

3 Integrtion by prts is then usully formulted by rerrnging slightly s follows: g(z)h(z) b b g(z)h (z)dz = b g (z)h(z)dz. Notice tht our pyment identity looks like the left-hnd side of this lst formul with g(z) = x(z) nd h(z) = z. Therefore p(v) = v zx (z)dz. This ltter formul is sometimes more convenient thn the former. We now mke concrete connection between the derivtive of the revenue curve nd the expected revenue of mechnism. Definition 4.2 The virtul vlue of n gent with vlue v F is: 1 φ(v) = B (v) f(v) = v 1 F(v) f(v). Lemm 4.3 For lloction rule x( ) nd v F, E[p(v)] = E[φ(v)x(v)]. Proof: Consider the following nlysis, where we use the definition of expecttion, swp the order of integrtion, use the definition of the cumultive distribution function, use the definition of the revenue curve, integrte by prts, plug in B() = B( ) =, use the definition of expecttion, nd finlly use the definition of virtul vlue. E v [p(v)] = = = = v zx (z) zx (z)dzf(v)dv z f(v)dvdz zx (z)(1 F(z))dz B(z)x (z)dz = B(z)x(z) B = (z) x(z)f(z)dz f(z) [ ] = E B (v) v x(v) f(v) = E v [φ(v)x(z)]. B (z)x(z)dz It should be no surprise, s shown by the bove lemm, tht the expected revenue cn be written entirely s function of the lloction rule. Wht is perhps surprising is tht this expected revenue is just the expecttion of the product of the lloction rule with the so-clled virtul vlue of the gent. 1 As we will discuss lter B (v)/f(v) is exctly the derivtive of the revenue curve in probbility spce. 65

4 Definition 4.4 The virtul surplus of outcome x nd vlues v is: Surplus(φ(v),x) = φ i(v i )x i c(x). i The following theorem follows from Lemm 4.3 nd linerity of expecttion. Theorem 4.5 The expected revenue of mechnism is equl to its expected virtul surplus, ] E v [ i φ i(v i )x i (v) c(x(v)). 4.3 Optiml Mechnisms nd Regulr Distributions We now derive the optiml mechnism for profit. To do this we gin wlk through stndrd pproch in mechnism design. We completely relx the incentive constrints nd sk nd solve the remining non-gme-theoretic optimiztion question. Since expected profit equls expected virtul surplus, this non-gme-theoretic optimiztion question is to optimize virtul surplus. We then verify tht this lgorithmic solution does not violte the incentive constrints (under some conditions). We conclude tht (under these conditions) the resulting mechnism is optiml. The the non-gme-theoretic optimiztion problem of mximizing virtul surplus is tht of finding x to mximize Surplus(φ(v),x) = i φ i(v i )x i c(x). 2 Let OPT gin be the surplus mximizing lgorithm. We will cre bout both the lloction tht OPT(φ(v)) selects, i.e., rgmx x Surplus(φ(v),x) nd its virtul surplus mx x Surplus(φ(v),x). Where it is unmbiguous we will use nottion OPT(φ(v)) to denote either of these quntities. Recll tht the formultion of OPT hs no mention of the incentive constrints. We know from the BIC chrcteriztion tht the lloction rule of ny BIC is monotone. Thus, the mechnism design problem of finding BIC mechnism to mximize virtul surplus hs n dded monotonicity constrint. Even though we did not impose monotonicity constrint on OPT, if the virtul vlution functions φ i ( ) re monotone, OPT(φ(v)) is monotone. Definition 4.6 Distribution F is regulr if φ( ) is non-decresing. Lemm 4.7 For ech gent i nd ny vlues of other gents v i, if F i is regulr then i s lloction rule from OPT on virtul vlues is monotone in i s vlue v i. Proof: Recll from Lemm 3.6 tht mximizing surplus is monotone. Mening, if we find x to mximize Surplus(v,x) then x i (v i, v i ) is monotone in v i. Therefore x i (φ i (v i ), φ i (v i )) is monotone in φ i (v i ), i.e., incresing φ i (v i ) does decrese x i. By the regulrity ssumption on F i, φ i (v i ) is monotone in v i. Therefore, incresing v i cnnot decrese φ i (v i ) which cnnot decrese x i. 2 Here the shorthnd nottion φ(v) = (φ 1 (v 1 ),..., φ n (v n )). 66

5 Since OPT on virtul vlues is monotone for ech gent nd ny vlues of other gents it stisfies our strongest incentive constrint. With the pproprite pyments (i.e., the criticl vlues ) truthtelling is dominnt strtegy equilibrium (recll Corollry 2.8). The resulting mechnism is due to Roger Myerson. Mechnism 4.1 The Myerson mechnism for regulr distributions is: 1. Solicit nd ccept seled bids b, 2. (x,p ) VCG(φ(b)), nd 3. for ech i, p i φ 1 i (p i ). Notice tht the pyments p clculted cn be viewed s the following. VCG on virtul vlues outputs virtul prices p. These correspond to the minimum virtul vlue n gent must hve to win. The Myerson mechnism then pplies the inverse virtul vlution function to determine the minimum vlue sid gent must hve to win. 3 Theorem 4.8 The Myerson mechnism for regulr distributions is IC. Corollry 4.9 The Myerson mechnism for regulr distributions mximizes expected revenue. It is quite useful to view this result s reduction from the problem of profit mximiztion to the problem of surplus mximiztion. As bove, we used the VCG mechnism to construct Myerson s mechnism. This generl reduction pplies to IC worst-cse surplus pproximtion mechnisms s well. Theorem 4.1 For ny IC mechnism M tht gives β-pproximtion to the optiml socil surplus, M(φ( )), with pyments computed vi inverse virtul vlutions, is β- pproximtion mechnism to the optiml expected profit. Notice tht there is not similr sttement for BIC mechnisms nd Byesin pproximtion s BIC nd Byesin pproximtion requires the input to the mechnism be v F. The distribution of virtul vlues is will not be the sme s the distribution of vlues nd therefore there is no implied gurntee of monotonicity nor expected revenue of the mechnism tht results from the bove construction. 4.4 Single-item Auctions The bove description of profit-optiml mechnisms does not offer much in the wy of intuition. To get clerer picture, we consider optiml mechnisms the specil cse of single-item 3 Assuming virtul vlutions re strictly non-decresing then the inverse virtul vlutions re well defined. We defer discussion of the non-strict cse to the subsequent section on irregulr distributions. 67

6 uctions, i.e., settings where fesible outcomes serve t most one gent. So wht is the mechnism tht optimizes virtul surplus for single-item settings? First notice tht virtul vlues cn be negtive. Consider the uniform distribution U[, 1] where F(z) = z nd f(z) = 1. Here φ(v) = v 1 F(z) = 2v 1. Thus, φ() = 1. If our gol f(z) is to optimize virtul surplus we clerly do not wnt to llocte to ny gent with negtive virtul vlue. Recll tht virtul vlues re the negtive derivtive of the revenue curve (normlized by the density function) nd our nlysis of the single-gent setting lredy suggested tht we should not llocte to n gent for whom this quntity is negtive. Second notice tht mong the gents with positive virtul vlues the virtul surplus is mximized by llocting to the one with the highest virtul vlue. Conclude the following corollry. Corollry 4.11 For regulr distributions, the optiml single-item uction lloctes to the gent with the highest non-negtive virtul vlution. As virtul vlutions re the negtive derivtive of the revenue curve (normlized by the density function) this mechnism sys to llocte to the gent whose revenue curve is the steepest t their vlue. The cse where the gents re independent nd identiclly distributed is of specil interest. Here the gent with the highest positive virtul vlue is lso the one with the highest vlue (s the virtul vlution functions re identicl). An gent s virtul vlue is nonnegtive when their vlue is t lest φ 1 (). Wht uction lloctes to the gent with the highest vlue tht is t lest φ 1 ()? It is the second price uction with reserve φ 1 ()! Corollry 4.12 For regulr i.i.d. distributions, F, the second-price uction with reserve φ 1 () is the single-item uction with the highest expected revenue. We conclude by returning to our two gent U[, 1] exmple. As we hve clculted, φ(v) = 2v 1; therefore, φ 1 () = 1/2. The second-price uction with reserve price 1/2 hs the optiml expected revenue. Our previous clcultion showed tht this revenue ws 5/12. While this uction is optiml mong BIC uctions, which is the clss of mechnisms we restricted our ttention to, the reveltion principle implies tht no uction hs BNE with higher expected revenue. Therefore, we conclude tht in very strong sense, tht the second price uction with reserve price mximizes revenue. 4.5 Irregulr Distributions nd Ironed Virtul Vlues We gin turn our ttention to the cse where the non-gme-theoretic optimiztion problem is not itself inherently monotone. An irregulr distribution is one for which the virtul vlution functions re not monotone non-decresing, i.e., where higher vlue might result in lower virtul vlue. Clerly OPT(φ( )) is non-monotone for such settings. 68

7 4.5.1 Quntile Spce In Chpter 3 we sw tht monotonizing non-monotone lloction rule ws more intuitive in quntile spce thn vlue spce. We will refer to the ex nte probbility t which n gent with vlue v F ccepts n offered price s the quntile of the price. As before, we will ssume tht the distribution function is continuous nd therefore the distribution function F(z) hs unique inverse, denoted F 1 (z). The trnsformtion from vlue spce into quntile spce is specified by the formul q = 1 F(v). In the preceding section we wrote the expected pyment s function of the lloction rule nd the revenue curve. Define y(q) = x(f 1 (1 q)) s the lloction rule in quntile spce. Define R(q) = q (F 1 (1 q)) = B(F 1 (1 q)) s the revenue curve in quntile spce. Notice tht incresing quntile corresponds to decresing vlue therefore the derivitives in quntile nd vlue spce hve oppposite sign. The derivtive of the revenue curve in quntile spce is exctly the negtive derivtive of the revenue curve in vlue spce normlized by the density function, i.e., R (q) = B (v)/f(v) for q = 1 F(v). Likewise derivtive of the lloction rule in quntile spce is exctly the negtive derivtive of the lloction rule in vlue spce normilized by the density function, i.e., y (q) = x (v)/f(v) for q = 1 F(v). Lemm 4.13 For lloction rule y( ) nd v F, E v [p(v)] = E q [R (q)y(q)] = E q [R(q)y (q)]. This lemm follows directly from the proof of Lemm 4.3 where the second equlity follows from the definition of expecttion nd integrtion by prts. This lterntive view point is useful s it immeditely implies the following corollry. Corollry 4.14 For lloction rule y( ) nd vlues v 1 F 1 nd v 2 F 2, if R 1 (q) R 2 (q) for ll q then E v1 [p(v 1 )] E v2 [p(v 2 )]. Notice tht virtul vlutions re simply the derivitive of the revenue curve in quntile spce, i.e., φ(v) = R (q) for q = 1 F(v) (Definition 4.2) Ironed Revenue Curves Recll the discussion of non-monotonicity in Chpter 3; if we tret Alice the sme regrdless of her vlue when her quntile is on some intervl [, b] then we cn replce her exct virtul vlution with her verge virtul vlution on this intervl. Figure 4.1() depicts hypotheticl non-concve revenue curve; the corresponding virtul vlue function, i.e., its derivtive, is depicted in Figure 4.1(c). Figure 4.1(d) shows Alice s virtul vlue verged on [, b]. Finlly, Figure 4.1(b) shows the resulting revenue curve. Notice tht the constnt virtul vlution over [, b] results in liner revenue curve, specificlly, the line segment connecting (, R()) to (b, R(b)). This process of treting Alice the sme on n intervl to fltten the virtul vlution function is known s ironing. It should be intuitively cler tht if we restrict ourselves to lloction rules tht tret Alice the sme on pproprite subintervls of quntile spce we cn construct n effective 69

8 1 1 1 b 1 1 () Revenue curve R(q). (b) Revenue curve R(q) ironed on [, b] (c) Virtul vlues R (q). 1 b 1 (d) Virtul vlues R (q) ironed on [, b]. Figure 4.1: On the left is the revenue curve R(q) nd virtul vlutions R (q) in quntile spce. On the right is the effective revenue curve nd virtul vlutions when ironed on [, b]. Though it is not necessry for understnding this exmple, this R( ) comes from bimodl distribution tht is U[, 2] with probbility 3/4 nd U[2, 8] with probbility 1/4. revenue curve R( ) equl to the concve hull of the ctul revenue curve R( ). This revenue curve is known s the ironed revenue curve nd its derivtive is the ironed virtul vlution function. Definition 4.15 for v F, the ironed revenue curve, R( ), is the concve hull of R( ) nd the ironed virtul vlution function is φ(v) = R (q) for q = 1 F(v). Notice the dvntge of R( ) over R( ) is two-fold. First, Lemm 4.14 suggests tht we cn get more revenue from R( ) thn from R( ). Second, R( ) is concve by definition, so ironed virtul vlutions re monotone, so ironed virtul surplus mximiztion results in monotone lloction rule, so with the pproprite pyment rule it is incentive comptible. It my perhps seem strnge tht for revenue curve R( ) somehow we re ble to get more revenue thn this revenue curve suggests, i.e., vi the ironed revenue curve R( ). In fct this is n rtifct of our formulic definition of the revenue curve s R(q) = qf 1 (1 q). Insted if look for optiml revenue curves see immeditely tht such curves re inherently concve. 7

9 A revenue curve should give the expected revenue s function of fixed probbility of sle. In our definition of R( ) we ssumed tht this comes from posting the price tht corresponds to this fixed probbility of sle. There is nother wy we cn sell with fixed probbility ˆq: pick some intervl [, b] with < ˆq < b nd consider the lloction rule 1 if q < yˆq ˆq (q) = if q [, b] b if b < q. Notice tht when Alice s quntile q is relized (i.e., drwn from the uniform distribution) then the probbility tht Alice is served by yˆq ( ) is 1 + ˆq (b ) = ˆq. The revenue b from such n lloction rule follows directly from Lemm It is R()+ ˆq (R(b) R()). b Notice tht this is exctly the vlue t ˆq on the line segment connecting (, R()) to (b, R(b)). Agin this cn be seen in Figure 4.1(b). Where R( ) is non-concve, this revenue cn be higher thn R(ˆq). Mening, if we wnt to sell to Alice with ex nte probbility ˆq there is better wy to do it thn offer price p = F 1 (1 ˆq); insted use lloction rule yˆq ( ), bove Optiml Mechnisms So fr we hve sid seen tht the ironed revenue curve domintes the revenue curve. Furthermore, if our lloction rule trets gents the sme in the ironed intervls then its virtul surplus equls its ironed virtul surplus. However, we still need to rgue tht lloction rules restricted in this mnner re indeed optiml. Therefore, we need to relte the expected pyment to R ( ) for ny (unrestricted) monotone lloction rule y( ). Lemm 4.16 For ny monotone lloction rule y( ) nd v F, Proof: E v [p(v)] = E q [ R (q)y(q) ] + E q [( R(q) R(q) ) y (q) ]. E v [p(v)] = E q [R (q)y(q)] + E q [ R (q)y(q) ] E q [ R (q)y(q) ] = E q [ R (q)y(q) ] E q [( R (q) R (q) ) y(q) ] = E q [ R (q)y(q) ] + E q [( R(q) R(q) ) y (q) ]. This lemm gives concrete suggestion for how to proceed. Pyment is equl to the expected ironed virtul vlue (first term) plus second term. Inspecting the second term, ( R(q) R(q) ) y (q), more closely, notice tht the difference in the revenue curves is nonnegtive, s R( ) is the concve hull of R( ); nd the derivtive of the lloction rule is nonpositive, s the lloction rule is monotone decresing in quntile. Therefore, the second term is non-positive nd mximizing it is equivlent to minimizing its mgnitude. As we see in the lemm below, if we ignore the second term nd mximize the first term then the second term will be zero, nd therefore it is lso mximized. We cn conclude tht mximizing ironed virtul surplus is optiml. 71

10 Lemm 4.17 For v F nd monotone lloction rule y( ) stisfying R (q) = y ( ) = then, E v [p(v)] = E q [ R (q)y(q) ]. Proof: We must rgue tht the ssumption on the lloction rule implies tht the second term of Lemm 4.16, E q [ [ R(q) R(q)]y (q) ], is zero. Notice tht R(q) R(q) > implies tht R( ) t q is on line segment connecting points < q < b on R(q). Therefore, R (q) is constnt, nd R (q), the second derivtive of the ironed revenue curve, is zero. Our ssumption then implies tht y (q) =. We conclude tht if the first multiplicnd of the expecttion is non-zero, then the second is zero; therefore, the expecttion is lwys identiclly zero. Consider the optimiztion function given by OPT( φ(v)), i.e., optimizing the ironed virtul surplus. Ech φ i ( ) is monotone since it is the derivtive of concve function. This is good news becuse it mens we cn construct n incentive comptible mechnism from OPT( φ( )). Referring bck to Lemm 4.16, by definition we hve mximized the first term (i.e., the ironed virtul surplus). Coincidentlly, we hve lso mximized the second term. To see this, notice tht the input to the optimiztion is ironed virtul vlues; therefore, the resulting lloction rule must be constnt on intervls of vlue (or quntile) where the ironed virtul vlues re constnt. Therefore, the ssumption of Lemm 4.17 is stisfied nd the second term (of Lemm 4.16) is zero, i.e., t its mximum. We conclude tht lloction rule of OPT( φ( )) mximizes expected profit. This rgument proves the following lemm. Lemm 4.18 For ny distribution F nd ny generl single-dimensionl gent setting, OPT( φ( )) mximizes virtul surplus subject to monotonicity. Since OPT on ironed virtul vlues is monotone for ech gent nd ll vlues of other gents, it stisfies our strongest incentive constrint. With the pproprite pyments (i.e., the criticl vlues ) truthtelling is dominnt strtegy equilibrium (recll Corollry 2.8). Mechnism 4.2 The Myerson mechnism, Mye F, for product distribution F is: 1. Solicit nd ccept seled bids b, 2. (x,p ) VCG( φ(b)), nd 3. clculte pyments for ech gent from the pyment identity. Unlike VCG nd the Myerson mechnism for regulr distirbutions (with strictly incresing virtul vlution functions) where the continuity ssumption on the distribution implies tht there is never tie, the Myerson mechnism for irregulr distributions my require tie-breking policy. Tie breking cn be done rbitrrily (s long s it is not function of the gents vlues). Common tie-breking rules re lexicogrphicl nd rndom. Lexicogrphicl tie breking will fvor sets of gents with higher indices. Rndom tie breking tkes the lexocogrphicl ordering on rndom permuttion of the gent indices. The rndomized tie-breking rule is nice becuse it is symmetric. 72

11 v 4 b v 3 v 2 v 1 v 4 b v 3 v 2 v 1 () Unique highest ironed virtul vlue. (b) Non-unique highest ironed virtul vlue. Figure 4.2: The ironed virtul vlution funciton φ(v) under two reliztions of gent vlues depicting both cse tht where the highest ironed virtul vlue is unique nd the cse were it is not unique. Theorem 4.19 The Myerson mechnism is IC. Corollry 4.2 The Myerson mechnism mximizes expected revenue. Like in the regulr cse, it is quite useful to view this result s reduction from the problem of profit mximiztion to the problem of surplus mximiztion. As bove, we used the VCG mechnism to construct the Myerson mechnism. This generl reduction pplies to IC worst-cse surplus pproximtion mechnisms s well. Theorem 4.21 For ny IC mechnism M tht gives β-pproximtion to the optiml socil surplus, M( φ( )), with pproprite pyments, is β-pproximtion mechnism to the optiml expected profit Single-item Auctions We consider the specil cse of single-item uctions to get clerer picture of exctly wht this optiml mechnism is in the cse of i.i.d. irregulr distributions. Figure 4.2 depicts hypotheticl ironed virtul vlution function. Instntiting the gents vlues corresponds to picking points on the x-xis. The gents ironed virtul vlutions cn then be red off the plot. The Myerson uction ssigns the item to the gent with the highest ironed virtul vlue. If there is tie, it picks rndom tied gent to win. Figure 4.2() depicts reliztion of vlues for n = 4 gents where the highest ironed virtul vlution is unique. Wht does Myerson do here? Myerson lloctes the item to this gent, i.e., gent 1 in the figure. Figure 4.2(b) depicts second reliztion of vlues where the highest ironed virtul vlution is not unique. In this setting Myerson, we will ssume, breks ties by picking rndom tied gent s the winner, i.e., one of gents 1, 2, nd 3 in the figure. In generl when there is k-gent tie for the highest ironed virtul vlution then ech tied gent wins with probbility 1/k. We now clculte the pyments. Consider the cse where there is unique highest ironed virtul vlue. The gent with this ironed virtul vlue wins. To clculte their IC pyment 73

12 1 1 p 1 b () < v i v 1 p 1 b v 1 (b) b v i Figure 4.3: The lloction (blck line) nd pyment rule (gry region) for gent 1 given v i nd the ironed virtul vlution function from Figure 4.2. we need to consider gent i s lloction rule for fixed vlues v i of the other gents. Consider gin the exmple in Figure 4.2() nd imgine the probbility we llocte to gent 1 s function of v 1. This is 1 if z > x i (v i, z) = 1/k if z [b, ] if z < b. when v i hs k 1 gents in [b, ] tied for the highest ironed virtul vlution. The 1/k probbility of winning for z [b, ] rises from our nlysis of wht hppens when in k-gent tie. Figure 4.3() depicts the lloction nd rule pyment of this gent. When gent 1 hs the unique highest ironed virtul vlue, i.e., v 1 > then clerly p 1 = ( b)/k. When gent 1 is tied for the highest ironed virtul vlue with k 1 other gents, s depicted in Figure 4.3(b), their expected pyment is p 1 = b/k. Of course, x 1 = 1/k so such pyment cn be implemented by chrging b to the tied gent tht wins. 4.6 Notes The optiml single-item uction ws derived by Myerson [18]. Its generliztion to singledimensionl gent settings is n obvious extension. The reltionship between optiml uctions, revenue curves, nd mrginl revenue (equivlent to virtul vlues) is due to Bulow nd Roberts [8]. 74